Integrand size = 14, antiderivative size = 41 \[ \int (c+d x) \cot ^2(a+b x) \, dx=-c x-\frac {d x^2}{2}-\frac {(c+d x) \cot (a+b x)}{b}+\frac {d \log (\sin (a+b x))}{b^2} \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3801, 3556} \[ \int (c+d x) \cot ^2(a+b x) \, dx=\frac {d \log (\sin (a+b x))}{b^2}-\frac {(c+d x) \cot (a+b x)}{b}-c x-\frac {d x^2}{2} \]
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Rule 3556
Rule 3801
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x) \cot (a+b x)}{b}+\frac {d \int \cot (a+b x) \, dx}{b}-\int (c+d x) \, dx \\ & = -c x-\frac {d x^2}{2}-\frac {(c+d x) \cot (a+b x)}{b}+\frac {d \log (\sin (a+b x))}{b^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.59 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.00 \[ \int (c+d x) \cot ^2(a+b x) \, dx=-\frac {c \cot (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(a+b x)\right )}{b}+\frac {d \log (\sin (a+b x))}{b^2}-\frac {d x \csc (a) (2 \cos (a)+b x \sin (a))}{2 b}+\frac {d x \csc (a) \csc (a+b x) \sin (b x)}{b} \]
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Time = 1.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.56
method | result | size |
default | \(-\frac {d \,x^{2}}{2}-c x +\frac {\frac {d a \cot \left (x b +a \right )}{b}-c \cot \left (x b +a \right )+\frac {d \left (-\left (x b +a \right ) \cot \left (x b +a \right )+\ln \left (\sin \left (x b +a \right )\right )\right )}{b}}{b}\) | \(64\) |
parallelrisch | \(\frac {-d \,x^{2} b^{2}-2 \cot \left (x b +a \right ) d x b -2 c x \,b^{2}-2 \cot \left (x b +a \right ) c b +2 d \ln \left (\tan \left (x b +a \right )\right )-d \ln \left (\sec \left (x b +a \right )^{2}\right )}{2 b^{2}}\) | \(66\) |
risch | \(-\frac {d \,x^{2}}{2}-c x -\frac {2 i d x}{b}-\frac {2 i d a}{b^{2}}-\frac {2 i \left (d x +c \right )}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}+\frac {d \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{b^{2}}\) | \(69\) |
norman | \(\frac {-\frac {c}{b}-c x \tan \left (x b +a \right )-\frac {d x}{b}-\frac {d \,x^{2} \tan \left (x b +a \right )}{2}}{\tan \left (x b +a \right )}+\frac {d \ln \left (\tan \left (x b +a \right )\right )}{b^{2}}-\frac {d \ln \left (1+\tan \left (x b +a \right )^{2}\right )}{2 b^{2}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (39) = 78\).
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.37 \[ \int (c+d x) \cot ^2(a+b x) \, dx=-\frac {2 \, b d x - d \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) + 2 \, b c + 2 \, {\left (b d x + b c\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b^{2} d x^{2} + 2 \, b^{2} c x\right )} \sin \left (2 \, b x + 2 \, a\right )}{2 \, b^{2} \sin \left (2 \, b x + 2 \, a\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (36) = 72\).
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.54 \[ \int (c+d x) \cot ^2(a+b x) \, dx=\begin {cases} \tilde {\infty } \left (c x + \frac {d x^{2}}{2}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \cot ^{2}{\left (a \right )} & \text {for}\: b = 0 \\\tilde {\infty } \left (c x + \frac {d x^{2}}{2}\right ) & \text {for}\: a = - b x \\- c x - \frac {d x^{2}}{2} - \frac {c}{b \tan {\left (a + b x \right )}} - \frac {d x}{b \tan {\left (a + b x \right )}} - \frac {d \log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b^{2}} + \frac {d \log {\left (\tan {\left (a + b x \right )} \right )}}{b^{2}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (39) = 78\).
Time = 0.36 (sec) , antiderivative size = 292, normalized size of antiderivative = 7.12 \[ \int (c+d x) \cot ^2(a+b x) \, dx=-\frac {2 \, {\left (b x + a + \frac {1}{\tan \left (b x + a\right )}\right )} c - \frac {2 \, {\left (b x + a + \frac {1}{\tan \left (b x + a\right )}\right )} a d}{b} + \frac {{\left ({\left (b x + a\right )}^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b x + a\right )}^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, {\left (b x + a\right )}^{2} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b x + a\right )}^{2} - {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) + 4 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} b}}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1151 vs. \(2 (39) = 78\).
Time = 0.55 (sec) , antiderivative size = 1151, normalized size of antiderivative = 28.07 \[ \int (c+d x) \cot ^2(a+b x) \, dx=\text {Too large to display} \]
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Time = 22.93 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.63 \[ \int (c+d x) \cot ^2(a+b x) \, dx=-\frac {d\,x^2}{2}+\frac {d\,\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b^2}-\frac {\left (c+d\,x\right )\,2{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}-\frac {x\,\left (b\,c+d\,2{}\mathrm {i}\right )}{b} \]
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